356 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



and at every surface of discontinuity, 



1 f 8F 8F 



\9) > " = I A*l J~ /^2 "" 



These equations will henceforth be known as the generalized 

 Poisson's equations. They give us the law of distribution of force 

 in the differential form, and contain the forms heretofore used as a 

 special case, obtained by putting //, = 1. 



The form of the integral expressing the energy is the same 

 as that of the integral J of 165 (i). All the properties of 

 the integral J are accordingly possessed by the integral Wf. In 

 particular it follows that if the potential is given at certain 

 surfaces the condition that the energy shall be a minimum re- 

 quires that in the space between 



(10) 1^9F) + 1^3F) + 1LL F ) = 



and on surfaces of discontinuity 



8F 8F 



We may call the problem of finding a function that shall 

 satisfy these differential equations, and take the required surface- 

 values, the generalized Dirichlet's Problem. The function F may 

 be called quasi-harmonic. 



It is evident, as in 86, that the solution of the problem, 

 if there be any, is unique. 



We have heretofore said nothing regarding the localization of 

 the energy of a distribution, which we have represented either by 

 an integral W$ throughout the acting distribution, or by an integral 

 Wf, which is expressed in terms of the field at all points of space. 

 Whereas both representations are equivalent mathematically, it is 

 a fundamental point in Maxwell's theory to regard the energy as 

 localized in the medium wherever a field exists. 



181. Induction. If we define a vector g by the equations 

 = g cos (ga?) = -p t 



(12) |t r g 



8F 



3 = g COS (g*) = - ft, 



